1. **State the problem:** Solve the equation $X(x-4) = (3-x)(x+3)$ for $x$. 2. **Write down the equation:** $$X(x-4) = (3-x)(x+3)$$ 3. **Expand the right side:** $$(3-x)(x+3) = 3x + 9 - x^2 - 3x = 9 - x^2$$ 4. **Rewrite the equation:** $$X(x-4) = 9 - x^2$$ 5. **Distribute $X$ on the left side:** $$Xx - 4X = 9 - x^2$$ 6. **Bring all terms to one side to set equation to zero:** $$Xx - 4X - 9 + x^2 = 0$$ 7. **Rearrange terms:** $$x^2 + Xx - 4X - 9 = 0$$ 8. **This is a quadratic equation in $x$: $$x^2 + Xx - (4X + 9) = 0$$** 9. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=X$, and $c=-(4X+9)$. 10. **Calculate the discriminant:** $$\Delta = X^2 - 4(1)(-(4X+9)) = X^2 + 16X + 36$$ 11. **Write the solutions:** $$x = \frac{-X \pm \sqrt{X^2 + 16X + 36}}{2}$$ **Final answer:** $$\boxed{x = \frac{-X \pm \sqrt{X^2 + 16X + 36}}{2}}$$ This gives the two possible values of $x$ depending on the sign chosen in the quadratic formula.